Given that a natural language such as English contains, at any given time, a finite number of words, any comprehensive list of definitions must either be circular or rely upon primitive notions.

Given also a measure on set, then, sometimes also denoted or, has as its vectors equivalence classes of measurable functions whose absolute value's-th power has finite integral, that is, functions for which one has

Given a finite dimensional real quadratic space with quadratic form, the geometric algebra for this quadratic space is the Clifford algebra Cℓ ( V, Q ).

Given a basis of a vector space, every element of the vector space can be expressed uniquely as a finite linear combination of basis vectors.

Given an arbitrary group G, there is a related profinite group G < sup >^</ sup >, the profinite completion of G. It is defined as the inverse limit of the groups G / N, where N runs through the normal subgroups in G of finite index ( these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between the quotients ).

Given a finite presentation P =

Given a Hilbert space L < sup > 2 </ sup >( m ), m being a finite measure, the inner product < ·, · > gives rise to a positive functional φ by

* Given an infinite string where each character is chosen uniformly at random, any given finite string almost surely occurs as a substring at some position.

* Given an infinite sequence of infinite strings, where each character of each string is chosen uniformly at random, any given finite string almost surely occurs as a prefix of one of these strings.

* Given a category C with finite coproducts, a cogroup object is an object G of C together with a " comultiplication " m: G → G G, a " coidentity " e: G → 0, and a " coinversion " inv: G → G, which satisfy the dual versions of the axioms for group objects.

Given a vector space V over a field K, the span of a set S ( not necessarily finite ) is defined to be the intersection W of all subspaces of V which contain S. W is referred to as the subspace spanned by S, or by the vectors in S. Conversely, S is called a spanning set of W.

Given this, it is quite natural and convenient to designate a general sequence a < sub > n </ sub > by by the formal expression, even though the latter is not an expression formed by the operations of addition and multiplication defined above ( from which only finite sums can be constructed ).

Given the rules of any two-person game with a finite number of positions, one can always trivially construct a minimax algorithm that would exhaustively traverse the game tree.

Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.

Given two fractional ideals I and J, one defines their product IJ as the set of all finite sums: the product IJ is again a fractional ideal.

Given the finite supply of natural resources at any specific cost and location, agriculture that is inefficient or damaging to needed resources may eventually exhaust the available resources or the ability to afford and acquire them.

Given a function w on U × Y, with finite integral of its modulus for any input function u and initial state x ( 0 ) over any finite time t, called the " supply rate ", a system is said to be dissipative if there exist a continuous nonnegative function V ( x ), with x ( 0 ) = 0, called the storage function, such that for any input u and initial state x ( 0 ) the difference V ( x ( t )) − V ( x ( 0 )) does not exceed the integral of the supply over ( 0, t ) for any t ( dissipation inequality ).

Given that finite fields are discrete in nature, and topology speaks only about the continuous, the detailed formulation of Weil ( based on working out some examples ) was striking and novel.

Given a finite set

Given a finite observation set S, one can simply select the measure for all.

For a finite group G, the left regular representation λ ( over a field K ) is a linear representation on the K-vector space V whose basis is the elements of G. Given g ∈ G, λ ( g ) is the linear map determined by its action on the basis by left translation by g, i. e.

Given two column vectors and of random variables with finite second moments, one may define the cross-covariance to be the matrix whose entry is the covariance.

Given an n-dimensional formal group law F over R and a commutative R-algebra S, we can form a group F ( S ) whose underlying set is N < sup > n </ sup > where N is the set of nilpotent elements of S. The product is given by using F to multiply elements of N < sup > n </ sup >; the point is that all the formal power series now converge because they are being applied to nilpotent elements, so there are only a finite number of nonzero terms.

Given a base scheme S, an algebraic torus over S is defined to be a group scheme over S that is fpqc locally isomorphic to a finite product of multiplicative groups.

: Given any family of nonempty sets, their Cartesian product is a nonempty set.

Given Ealdred's known support of Godwin's family, John of Worcester is probably correct.

Given that the bulk of family chores fell to women, and that they were five times more likely than men to head a single-parent family, the shortcomings of Finland's child day-care system affected women more than it did men.

Given our political differences, people are sometimes surprised how close Ronnie and I have been to the Kennedy family.

Given the fact that many of the royal depictions of Akhenaten often featured such an elongated head, it is likely an exaggeration of a family trait, rather than a distinct abnormality.

Given two affine spaces and, over the same field, a function is an affine map if and only if for every family of weighted points in such that

Given a parameterized family of probability density functions ( or probability mass functions in the case of discrete distributions )

Given the harsh conditions of the retreat, the infant was left with a local family ( Two Europeans retracing the Long March route in 2003 met a woman in rural Yunnan province, said by local officials to be Mao and He Zizhen's long-lost daughter ).

Given a topological space X, a base for the closed sets of X is a family of closed sets F such that any closed set A is an intersection of members of F.

Given the broad expanse of the Malvasia family, generalizations about the Malvasia wine are difficult to pin point.

Given Harry Evans Sr .' s destructive character, Mary Evans would often leave home with her sons to nearby Somerville, to stay with her sister Justine and the Epps family.

Dr. Tennison states: " Given the account of Elliot Paul, and given that Lead Belly witnessed boogie-woogie in 1899 in the Arklatex ; and given the North to South migration of the Thomas family ; and given the Texas & Pacific headquarters in Marshall in the early 1870s ; and given that Harrison County had the largest slave population in the state of Texas ; and given the fact that the best-documented and largest-scale turpentine camps in Texas did not occur until after 1900 in Southeast Texas, it is most probable that boogie-woogie spread from Northeast to Southeast Texas, rather than from Southeast to Northeast Texas, or by having developed diffusely with an even density over all of the Piney Woods of East Texas.

Given the sheer power of leverage the Rothschild family had at its disposal, this profit was an enormous sum.

* Given a family of continuous functions f < sub > n </ sub >: X → Y with pointwise limit f: X → Y.

Given his father's naval career, where the family was domiciled is uncertain.

Given this situation, Jose Antonio del Valle had to move his family to Guatemala City, where he hoped young Jose would be better formed.

Given this outlook, whatever the level of affection Whistler may have felt for his own mother, one finds an even more divergent use of the image in the Victorian era and later, especially in the United States, as an icon for motherhood, affection for parents, and " family values " in general.

Given a family of sets X < sub > i </ sub > the product is defined as

Given any set Y with a family of functions

Given the difficulties faced in Berlin, he decided to accept the offer and immediately moved to Göttingen with his family.

Given a manifold and a Lie algebra valued 1-form, over it, we can define a family of p-forms:

Given her mother's ambitions for her and the fact that Rudolf was married to Princess Stephanie of Belgium, her family and friends found this liaison to be not only foolish, but potentially socially compromising for the family as well.

* Given names often turned into family names when people were identified by their father's name.